In communication systems, bit error rate (BER) is a ratio of the number of bits, elements, characters, blocks, etc. incorrectly received to the total number of bits, elements, characters, blocks, etc. sent during a specified time interval. BER is one of the most fundamental and robust measurements of performance in a digital transmission system. Note, for high transmission rate communication systems, BER is estimated based on a signal bit rate determined over a measurement period. For example, to test a communication system at 10 Gb/s for a BER of 10−12, a test would require several hours. However, to test a communication system at 10 Gb/s for a BER of 10−15, the test would require months. Accordingly, BER is typically modeled based on computation techniques known in the art.
Optical communication systems, such as dense wavelength division-multiplexed (DWDM) systems, require repeated bit error rate (BER) computations for system optimization in a multi-dimensional parameter space. In order for any BER computation technique to become a practical engineering tool, it is highly desirable to reduce the computational (computer processing unit (CPU)) time to a few minutes or less for a single processor computer and to provide high computation accuracy at the same time. Despite dramatic progress in modeling wavelength division-multiplexed (WDM) fiber communication systems made in recent decades, computationally-efficient and accurate BER calculation still remains a challenge.
There are four state-of-the art computation techniques for calculating BER: 1) direct computation of BER using Monte Carlo simulations, 2) multi-canonical Monte Carlo simulation or importance sampling techniques, 3) indirect computation method (Q-factor technique), and 4) Karhunen-Loéve expansion technique. In the direct Monte-Carlo technique and the multi-canonical Monte Carlo (or importance sampling) technique, the BER is computed directly. In the Q-factor technique, the BER is computed indirectly in two steps. First, a Q-factor is computed as a cumulant of the signal statistics. Second, by assuming a pre-determined shape of the signal probability distribution function (pdf), the signal pdf is inferred and the BER is calculated for a given Q-factor. In the Karhunen-Loéve expansion technique, the BER is computed for each bit of a pseudo-random bit sequence (PRBS) individually and the final pdf is found by averaging over all the bits in the PRBS. The core element in all four aforementioned techniques is use of long pseudo-random bit sequences that are needed for adequate accuracy of the BER computations.
Directly computing the BER using Monte Carlo simulations is prohibitively inefficient. Use of the multi-canonical Monte Carlo simulation or importance sampling techniques drastically increases the computational efficiency. However, the later two approaches still require typically long (more than one hour) CPU time which is often too long for practical use. Drawbacks of the Q-factor approach are twofold. First, there is always a tradeoff between the Q-factor calculation accuracy itself and the pseudo-random bit sequence (PRBS) word length, which requires running fairly long PRBS words resulting in long CPU times. Second, fitting parameters, which depend on the modulation format, are required to couple the Q-factor with the BER. These fitting parameters are inferred from the BER measurements. However, once the modulation format (or bit rate, or the receiver type) is changed the fitting parameters have to be re-defined again.
If the real signal pdf derived from the first principles depending on the system parameters is known, the BER could be determined directly without the fitting parameters. It is the lack of knowledge of the actual pdf which makes the BER modeling so inefficient. Use of the Karhunen-Loéve expansion technique helps to significantly increase the BER computation accuracy. However, it still requires the split-step Monte Carlo simulations with at least a 27−1 PRBS length to account for the data pattern dependent nonlinear signal distortion leading to increased CPU time. A common drawback in all four abovementioned techniques is that they all need long PRBS for adequate accuracy of the BER computations which results in long CPU times.